🧬 biology

Tree-Structured Orthonormal Decomposition of the Aitchison Simplex

This paper introduces PolyILR, a novel method that constructs a canonical orthonormal basis for compositional data aligned with any tree topology, thereby preserving intrinsic Aitchison geometry while enabling stable, interpretable, and multiscale analysis of hierarchical structures in fields like microbiome and genomics.

Original authors: Daisuke Yamada, Qijun Zhang, Travis Pence, Barbara B. Bendlin, Federico Rey, Vikas Singh

Published 2026-06-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daisuke Yamada, Qijun Zhang, Travis Pence, Barbara B. Bendlin, Federico Rey, Vikas Singh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

The Big Picture: Sorting the "Relative" Puzzle

Imagine you are a chef trying to describe a soup. You can't just say, "There are 5 carrots and 3 potatoes," because the total amount of soup changes depending on how much water you add. What actually matters is the ratio: the balance between carrots and potatoes. In science, data like this (microbes in your gut, cell types in your blood, or chemical mixtures) is called compositional data.

The problem scientists face is that these ratios are tricky to analyze with standard math tools. Furthermore, these ingredients aren't random; they belong to a family tree. For example, in your gut, Streptococcus and Lactobacillus are cousins (both bacteria), but they are very different from a virus.

The Problem: The "Binary" Bottleneck

For a long time, the best tool to analyze these ratios was called ILR (Isometric Log-Ratio). Think of ILR as a translator that turns "soup ratios" into standard numbers a computer can understand.

However, this translator had a major flaw: it was designed for binary trees (where every family branch splits into exactly two). But nature is messy. In the real world, a family branch often splits into three, four, or more at once (like a grandmother having five children at once).

To use the old tool, scientists had to force these messy, multi-branch trees into neat, two-branch splits.

The Analogy: Imagine you have a family tree where one grandparent has five children. To use the old tool, you have to arbitrarily decide, "Okay, let's pretend the first two children are one group, and the other three are another." You then have to guess how to split the remaining three.
The Consequence: This "guessing" (called binarization) introduces artificial choices. If you change how you split the group, your scientific results change. It's like measuring a room with a ruler that you can stretch or shrink depending on your mood; the measurement isn't reliable.

The Solution: PolyILR (The "Poly" Translator)

The authors of this paper created a new method called PolyILR. The "Poly" stands for polytomous, meaning it handles trees with many branches naturally.

How it works (The Metaphor):
Imagine the family tree as a series of rooms in a house.

Old Method: If a room had 5 doors leading to other rooms, the old method forced you to build a fake wall to split the doors into two groups, then split those groups again. You ended up with a maze of fake walls that didn't exist in the original house.
PolyILR: This method respects the original architecture. If a room has 5 doors, it creates a special, custom-made "balance scale" for that specific room that can weigh all 5 doors against each other simultaneously.

The "Weighted" Magic:
The paper's key innovation is a "weighted local geometry."

Imagine one branch of the tree leads to 1 leaf (a single person), and another branch leads to 100 leaves (a huge family).
If you just compare them equally, the big family drowns out the single person.
PolyILR acts like a smart scale. It knows that the big family is "heavier" in terms of numbers, so it adjusts the weight of the comparison so that every single leaf (every single person) gets a fair voice in the final calculation.

Why This Matters: Stability and Clarity

The paper tested this new method on real-world data (microbes in the human body and different types of blood cells) and found two major benefits:

Stability (No More Guessing):
- With the old method, if you shuffled the order of the "fake splits," you got different "top 10" important ingredients every time. It was like flipping a coin to decide which family member matters most.
- With PolyILR, the results are stable. No matter how you run the analysis, the same biological comparisons pop up as important. The method is "canonical," meaning there is only one correct way to do it based on the tree structure provided.
Interpretability (Reading the Map):
- Because PolyILR respects the tree, every number it produces corresponds to a specific location on the tree.
- The Analogy: If the old method gave you a list of "Feature #42," you wouldn't know what that meant. PolyILR gives you a label like "The balance between Streptococcus and the rest of the Lactobacillus family." You can look at the tree and immediately understand what the number represents.

The "Softmax" Connection (A Side Note)

The paper also mentions a theoretical link to Softmax classifiers (a common tool in AI for sorting things into categories).

The Analogy: The authors realized that the math used to sort probabilities in AI (like deciding if an image is a cat or a dog) is secretly the same math used to sort soup ratios.
The Claim: They suggest that if you have a hierarchy of categories (like "Animal -> Mammal -> Dog"), you could use this new tree-based math to better understand how AI models make mistakes or learn, by looking at the "balance" between categories rather than just the final answer.

Summary

The Problem: Existing tools for analyzing relative data (like microbes or cell types) force complex family trees into simple two-way splits, creating arbitrary and unstable results.
The Solution: PolyILR is a new mathematical tool that handles complex, multi-branch trees directly without forcing artificial splits.
The Result: It produces stable, reliable, and easy-to-understand results where every number corresponds to a specific, meaningful comparison in the family tree. It works for microbiome data, cell biology, and potentially for understanding how AI learns hierarchies.

Technical Summary: Tree-Structured Orthonormal Decomposition of the Aitchison Simplex

1. Problem Statement

Compositional data—vectors of non-negative components summing to a constant (e.g., microbial abundances, cell type proportions)—reside on the Aitchison simplex. Standard analysis relies on Aitchison geometry, where inference is based on ratios rather than absolute values. The Isometric Log-Ratio (ILR) transform maps this simplex to a Euclidean tangent space, providing an orthonormal coordinate system.

However, existing ILR methods face a critical limitation when dealing with polytomous trees (nodes with more than two children), which are ubiquitous in real-world hierarchies like taxonomies, phylogenies, and ontologies.

Structural Incompatibility: While binary trees allow a one-to-one mapping between internal nodes and log-ratio contrasts (balances), polytomous nodes require $k-1$ orthogonal contrasts to capture the full local structure of $k$ children.
Arbitrary Choices: Current approaches often force polytomous trees into binary resolutions (e.g., PhILR), introducing artificial nodes and splits not present in the original hierarchy. This leads to coordinate systems that are non-canonical, unstable, and dependent on arbitrary binarization choices.
Incomplete Geometry: Existing methods either ignore the hierarchical structure or fail to provide a complete, canonical orthonormal basis that respects the full branching structure of arbitrary trees without sacrificing the isometric property of the Aitchison geometry.

2. Methodology: PolyILR

The authors propose PolyILR (Polytomous ILR), a canonical orthonormal decomposition of the Aitchison tangent space aligned with any rooted tree topology, including polytomous ones.

Core Construction:

Local Geometry at Internal Nodes: For each internal node $u$ with $k_u$ children, the method defines a local contrast subspace $S_u \subset \mathbb{R}^{k_u}$ consisting of vectors summing to zero. This subspace captures all relative comparisons among the $k_u$ descendant clades.
Weighted Inner Product: To ensure that local orthonormality extends to global orthonormality, the method equips $S_u$ with a weighted inner product:
$\langle h, h' \rangle_w = \sum_{r=1}^{k_u} \frac{h_r h'_r}{n_r}$
where $n_r$ is the number of leaves in the $r$ -th child's subtree. This weighting accounts for unequal subtree sizes, ensuring that spreading local vectors to global leaf vectors preserves orthogonality.
Canonical Basis Selection: The authors use Helmert contrasts as a deterministic starting point for the local basis. These are then orthonormalized under the weighted inner product $\langle \cdot, \cdot \rangle_w$ using the Gram-Schmidt process. This yields a unique local basis $\tilde{H}(u)$ given a fixed child ordering.
Global Spreading: Each local basis vector is "spread" to the global leaf space $\mathbb{R}^d$ . If a leaf $i$ descends from child $r$ of node $u$ , its value in the global vector is scaled by $1/n_r$ . This specific scaling ensures that the global inner product of two spread vectors equals their local weighted inner product.
Assembly: Collecting the spread vectors from all internal nodes forms the global basis matrix $V \in \mathbb{R}^{d \times (d-1)}$ .

Theoretical Properties:

Isometry: The transformation $\phi(x) = V^\top \log x$ is an isometry from the Aitchison simplex to $\mathbb{R}^{d-1}$ .
Contrast Property: $V^\top \mathbf{1} = 0$ , ensuring coordinates represent log-ratios.
Orthonormality: $V^\top V = I_{d-1}$ .
Canonicity: Given a fixed tree topology, leaf labels, and child ordering, the basis is unique. Furthermore, the tree topology can be recovered from the support structure of the basis columns.

3. Key Contributions

Canonical Decomposition: PolyILR provides the first complete, canonical orthonormal decomposition of the Aitchison tangent space for arbitrary tree topologies, resolving the incompatibility between Aitchison geometry and polytomous hierarchies.
Interpretability: Each coordinate in the transformed space corresponds bijectively to a specific node-contrast pair in the tree. This allows for direct interpretation of features as log-ratios between specific groups of leaves (clades) at any resolution.
Stability: By avoiding arbitrary binarization, PolyILR yields stable feature selection. In contrast, methods requiring binary resolution (like PhILR) produce unstable feature sets that vary significantly with different binarization choices.
Multiscale Inference: The construction enables structured hypothesis testing at multiple resolutions (e.g., by depth, subtree, or specific node), as coordinates at distinct nodes span orthogonal subspaces.
Theoretical Connection to Softmax: The authors establish a novel theoretical link between the Aitchison tangent space and the logit space of softmax classifiers. They show that the quotient of logit space under shift equivalence is isomorphic to the Aitchison tangent space, suggesting PolyILR coordinates can be used to analyze model errors and gradients in hierarchical classification tasks.

4. Experimental Results

The authors evaluated PolyILR on standard microbiome (HMP, cMD3) and single-cell (DISCO) benchmarks.

Representation Validity: PolyILR, PhILR, and CLR yielded identical performance for linear models (SVM, Logistic Regression), confirming the geometric equivalence (isometry). Random Forest performance varied modestly, consistent with sensitivity to axis choice.
Feature Stability: PolyILR demonstrated high stability in feature selection (Jaccard similarity of top- $K$ features across cross-validation folds), whereas PhILR collapsed to near-zero stability due to the instability of arbitrary binarization.
Interpretability: The top features identified by PolyILR corresponded to biologically meaningful contrasts consistent with existing literature (e.g., Streptococcus vs. Lactococcus in oral subsites; Prevotella vs. Bacteroides in westernization; myeloid vs. erythroid imbalances in leukemia).
Tree-Level Inference: Aggregating feature importance by tree depth, subtree, or node revealed that predictive signals concentrate at specific hierarchical levels (e.g., coarse-level contrasts for westernization; immune-cell-level contrasts for leukemia), aligning with known biological structures.
Robustness: Preliminary analysis showed that PolyILR remains stable under topological noise (nearest-neighbor interchanges), with only local changes to the basis vectors, unlike the global disruption caused by binarization.

5. Significance and Claims

The paper claims that PolyILR addresses a fundamental gap in compositional data analysis by providing a geometrically principled, canonical, and complete coordinate system for data organized by arbitrary trees.

Interpretability over Performance: The primary goal is the interpretability of the representation itself. Each coordinate is a specific, tree-grounded contrast, eliminating the need for post-hoc interpretation of anonymous indices.
Generalization: PolyILR generalizes PhILR (which is restricted to binary trees) to arbitrary topologies without sacrificing the isometric properties of the Aitchison geometry.
Broader Applicability: The connection to softmax classifiers suggests that PolyILR could serve as a tool for analyzing hierarchical probabilistic models, potentially localizing model errors within a class hierarchy.

The authors acknowledge limitations: PolyILR requires a known, fixed tree as input and does not inherently handle topological uncertainty (e.g., bootstrap support). Additionally, like all log-ratio methods, it requires zero-replacement preprocessing. The paper positions these as areas for future work rather than current failures.